×

Layer potentials for elastostatics and hydrostatics in curvilinear polygonal domains. (English) Zbl 0711.35041

A classical way to treat boundary value problems of elliptic type is that of transforming the system of partial differential equations into a system of integral equations. As it is known, these systems have regular kernels and their Fredholm indices are not always zero when the boundary data are of class \(L^ p\) with \(p\neq 0\). Here the question of their solutions is examined in detail for the operators of classical elastostatics and hydrostatics when the region is a polygon with curvilinear sides in \({\mathbb{R}}^ 2.\)
Unusually, the system of integral equations is not directly studied, but replaced by a system of pseudodifferential operators of Mellin’s type, which offer the advantage of highlighting the singular part of the kernel. It is demonstrated that the operators are regular for \(2\leq p<+\infty\), and that singularities may occur only for \(p<2.\)
The system of integral equations is subsequently written in detailed form in the case in which the domain is a plane sector of angle \(\theta\). Since this domain is infinite the previous results must be revised. For \(0\leq \theta \leq 2\pi\) there is a critical value \(\theta_{{\mathfrak a}}=257\circ 27'\) for which the system becomes singular, so confirming a result already known in plane elasticity under the name of “Carother’s paradox.”
Reviewer: P.Villaggio

MSC:

35J55 Systems of elliptic equations, boundary value problems (MSC2000)
45E05 Integral equations with kernels of Cauchy type
35S15 Boundary value problems for PDEs with pseudodifferential operators
74B10 Linear elasticity with initial stresses
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Martin Costabel, Singular integral operators on curves with corners, Integral Equations Operator Theory 3 (1980), no. 3, 323 – 349. · Zbl 0445.45025 · doi:10.1007/BF01701497
[2] B. E. J. Dahlberg, C. E. Kenig, and G. C. Verchota, Boundary value problems for the systems of elastostatics in Lipschitz domains, Duke Math. J. 57 (1988), no. 3, 795 – 818. · Zbl 0699.35073 · doi:10.1215/S0012-7094-88-05735-3
[3] Johannes Elschner, Asymptotics of solutions to pseudodifferential equations of Mellin type, Math. Nachr. 130 (1987), 267 – 305. · Zbl 0663.35100 · doi:10.1002/mana.19871300125
[4] E. B. Fabes, C. E. Kenig, and G. C. Verchota, The Dirichlet problem for the Stokes system on Lipschitz domains, Duke Math. J. 57 (1988), no. 3, 769 – 793. · Zbl 0685.35085 · doi:10.1215/S0012-7094-88-05734-1
[5] Математические вопросы динамики вязкойнесжимаемой жидкости, Государств. Издат. Физ.-Мат. Лит., Мосцощ, 1961 (Руссиан). О. А. Ладыженская, Тхе матхематицал тхеоры оф висцоус инцомпрессибле флощ, Ревисед Енглиш едитион. Транслатед фром тхе Руссиан бы Ричард А. Силверман, Гордон анд Бреач Сциенце Публишерс, Нещ Ыорк-Лондон, 1963.
[6] Jeff E. Lewis and Cesare Parenti, Pseudodifferential operators of Mellin type, Comm. Partial Differential Equations 8 (1983), no. 5, 477 – 544. · Zbl 0532.35085 · doi:10.1080/03605308308820276
[7] Samuel N. Karp and Frank C. Karal Jr., The elastic-field behavior in the neighborhood of a crack of arbitrary angle, Comm. Pure Appl. Math. 15 (1962), 413 – 421. · Zbl 0166.20705 · doi:10.1002/cpa.3160150404
[8] V. D. Kupradze, Potential methods in the theory of elasticity, Translated from the Russian by H. Gutfreund. Translation edited by I. Meroz, Israel Program for Scientific Translations, Jerusalem; Daniel Davey & Co., Inc., New York, 1965. · Zbl 0188.56901
[9] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. · Zbl 0207.13501
[10] D. Vasilopoulos, On the determination of higher order terms of singular elastic stress fields near corners, Numer. Math. 53 (1988), no. 1-2, 51 – 95. · Zbl 0649.73011 · doi:10.1007/BF01395878
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.